TY - JOUR

T1 - Implicit solutions with consistent additive and multiplicative components

AU - Areias, P.

AU - Rabczuk, T.

AU - Dias-Da-Costa, D.

AU - Pires, E. B.

N1 - Funding Information:
The authors gratefully acknowledge financing from the “ Fundação para a Ciência e a Tecnologia ” under the Project PTDC/EME-PME/108751 and the Program COMPETE FCOMP-01-0124-FEDER-010267 .

PY - 2012/9

Y1 - 2012/9

N2 - This work describes an algorithm and corresponding software for incorporating general nonlinear multiple-point equality constraints in a implicit sparse direct solver. It is shown that direct addressing of sparse matrices is possible in general circumstances, circumventing the traditional linear or binary search for introducing (generalized) constituents to a sparse matrix. Nested and arbitrarily interconnected multiple-point constraints are introduced by processing of multiplicative constituents with a built-in topological ordering of the resulting directed graph. A classification of discretization methods is performed and some re-classified problems are described and solved under this proposed perspective. The dependence relations between solution methods, algorithms and constituents becomes apparent. Fracture algorithms can be naturally casted in this framework. Solutions based on control equations are also directly incorporated as equality constraints. We show that arbitrary constituents can be used as long as the resulting directed graph is acyclic. It is also shown that graph partitions and orderings should be performed in the innermost part of the algorithm, a fact with some peculiar consequences. The core of our implicit code is described, specifically new algorithms for direct access of sparse matrices (by means of the clique structure) and general constituent processing. It is demonstrated that the graph structure of the second derivatives of the equality constraints are cliques (or pseudo-elements) and are naturally included as such. A complete algorithm is presented which allows a complete automation of equality constraints, avoiding the need of pre-sorting. Verification applications in four distinct areas are shown: single and multiple rigid body dynamics, solution control and computational fracture.

AB - This work describes an algorithm and corresponding software for incorporating general nonlinear multiple-point equality constraints in a implicit sparse direct solver. It is shown that direct addressing of sparse matrices is possible in general circumstances, circumventing the traditional linear or binary search for introducing (generalized) constituents to a sparse matrix. Nested and arbitrarily interconnected multiple-point constraints are introduced by processing of multiplicative constituents with a built-in topological ordering of the resulting directed graph. A classification of discretization methods is performed and some re-classified problems are described and solved under this proposed perspective. The dependence relations between solution methods, algorithms and constituents becomes apparent. Fracture algorithms can be naturally casted in this framework. Solutions based on control equations are also directly incorporated as equality constraints. We show that arbitrary constituents can be used as long as the resulting directed graph is acyclic. It is also shown that graph partitions and orderings should be performed in the innermost part of the algorithm, a fact with some peculiar consequences. The core of our implicit code is described, specifically new algorithms for direct access of sparse matrices (by means of the clique structure) and general constituent processing. It is demonstrated that the graph structure of the second derivatives of the equality constraints are cliques (or pseudo-elements) and are naturally included as such. A complete algorithm is presented which allows a complete automation of equality constraints, avoiding the need of pre-sorting. Verification applications in four distinct areas are shown: single and multiple rigid body dynamics, solution control and computational fracture.

KW - Computational fracture

KW - Multibody dynamics

KW - Multiple-point constraints

KW - Sparse matrices

UR - http://www.scopus.com/inward/record.url?scp=84859396563&partnerID=8YFLogxK

U2 - 10.1016/j.finel.2012.03.007

DO - 10.1016/j.finel.2012.03.007

M3 - Article

AN - SCOPUS:84859396563

SN - 0168-874X

VL - 57

SP - 15

EP - 31

JO - Finite Elements in Analysis and Design

JF - Finite Elements in Analysis and Design

ER -